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Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

The claim is that, by embedding SU(N)SU(N) Yang-Mills in a larger (nonunitary) theory, whose gauge groups is the supergroup SU(N|N)SU(N|N), spontaneously broken to SU(N)×SU(N)SU(N)\times SU(N), one can produce a gauge-invariant Pauli-Villars regulator, with which to implement the Exact RG. The latter theory, in turn, is what Okuda and Takyanagi argue is the world-volume theory of a stack of D-branes and ghost D-branes.

When the gauge symmetry is unbroken, the SU(N|M)SU(N|M) theory is equivalent to SU(N−M)SU(N-M), as far as computing gauge-invariant observables. In particular, there is a perfect cancellation of diagrams for N=MN=M.

Turning on a nonzero Higgs VEV (separating the D-branes from the ghost D-branes) provides a cutoff for the original SU(N)SU(N) theory. Above the scale of the Higgs VEV, you get zero; far below it, the “original” SU(N)SU(N) degrees of freedom decouple from the ghost SU(N)SU(N).

Evans et al propose and AdS/CFT geometry realization of this idea, with the hope of connecting, in a explicit way, the “holographic RG” (evolution in the radial coordinate of AdS) with the “exact RG” of Morris.

Anyway, Takuya Okuda is over there, fielding questions, so take advantage …

They argue for two propositions which “must” be true in any theory of quantum gravity, but which are not obvious, at all, from the point of view of low-energy effective field theory. They both concern theories in which there is an unbroken U(1)U(1) gauge symmetry in the low-energy theory.

First, they argue that there must exist charged particle(s) in the theory, whose charge-to-mass ratio exceeds1 the extremal bound on the charge-to-mass ratio of blackholes. A-priori, you could imagine that one could twiddle the masses and charges of fundamental particles arbitrarily. However, as they make clear, in the absence of particles which exceed the bound, charged blackholes cannot radiate away all their charge, and one is left with a large (possibly infinite) number of charged remnants.

This is a perfectly solid result, and one which can be understood quite clearly, once one takes account of blackholes and their evaporation. From it, they abstract away the slogan, “Gravity is the weakest force.” Which leads them to their second conjecture.

The strength of the effective gravitational force grows like a power-law in the UV. The strength of gauge-interactions vary only logarithmically (growing in the UV for abelian gauge theories and falling in the UV for asymptotically-free nonabelian gauge theories). If we go to high enough energies, the gravitational force, therefore, comes to dominate, or would do so if the theory were not cut off.

If we ignore the slow logarithmic running of the gauge coupling, demanding that gauge-interactions dominate over gravitational ones puts a cutoff on effective field theory, not at MplM_{\text{pl}}, but at a lower scale, gMplg M_{\text{pl}}.

Now, it’s certainly true in all known string theories, 4D effective field theory breaks down below (often, well below) the 4D Planck scale. Indeed, as Arkani-Hamed knows well, the scale at which 4D effective field theory breaks down could be as low as several TeV. I firmly believe (along with the authors) that this is a general principle. But, to put a precise upper bound on the cutoff, at which 4D effective field theory must break down, does require taking account of the running of the gauge coupling.

To sharpen the conjecture, the authors assert that “gg” in the above formula is the low-energy value of the gauge coupling below the mass of the lightest charged particle. This is not directly related to the “high-energy” value of the coupling (close to the cutoff scale). The rate at which the coupling runs depends on massive charged species at intermediate scales. Not just the magnitude, but even the sign of the β\beta-function could change (if the abelian gauge theory is un-higgsed into a nonabelian one). So the scale at which gravity and gauge interactions become comparable in strength cannot be determined from low-energy data alone; it could be higher or lower than the “naïve” estimate of gIRMplg_{\text{IR}}M_{\pl}.

Indeed, in many string backgrounds, SU(2)×U(1)SU(2)\times U(1) is unbroken, and the quarks and leptons are massless. (In the same approximation, supersymmetry is, frequently, also unbroken.) In that case, the U(1)U(1) gauge coupling is driven all the way to zero in the IR2. But that does not mean that effective field theory has zero range of validity.

One can imagine a self-consistent bound of the form Λ=g(Λ)Mpl\Lambda = g(\Lambda) M_{\text{pl}}. That’s the form of the bound that they actually check in examples. (It’s g(Λ)g(\Lambda) that is directly related to gstg_{\text{st}}, not gIRg_{\text{IR}}.) In that form, as they verify, the bound holds3. But it’s not a form that depends solely on low-energy data.

1 In the BPS case, saturates the bound.

2 The fact that a massless electron causes the gauge coupling to flow to zero in the IR does not contradict the previous argument about the charge-to-mass ratio. Even though both mm and (log(m))−1/2(\log(m))^{-1/2} vanish as m→0m\to 0, the latter vanishes more slowly, so we preserve the fact that we have particles whose charge-to-mass ratio exceeds the extremal bound.

3 After some back-and-forth over email, Nima Arkani-Hamed agrees that this, rather than Λ=gIRMpl\Lambda = g_{\text{IR}}M_{\text{pl}}, is the bound. My argument is that it directly expresses the idea that Λ\Lambda is the scale at which gravity and gauge interactions become comparable in strength. Nima had a more sophisticated argument, involving the evaporation of magnetically-charged blackholes.

with the accesskeys for your site. To make them modifiable, I proposed a Javascript, which turns that definition-list into a template for editing the keybindings, and stores your modified keybindings for future visits.

But what about all those millions of sites, with accesskeys defined, which haven’t gone to this trouble? Since Gez Lemon inspired me to write the above Javascript, it was my turn to inspire Gez. Firefox users can install this Greasemonkey script which provides an (admittedly slightly crude) approximation to the same functionality on any site with accesskeys.

Without an author-supplied definition-list, Gez’s script needs to guess at the meaning of each accesskey. Usually, that works passably well. But, on some sites, the result can be downright comical. Obviously, it would be better if website authors went to the trouble of providing the aforementioned definition list. And with the aforementioned Javascript, all their users (not just those with Firefox and Greasemonkey installed) could customize those keybindings.

But still, it’s remarkable that, with a little clever scripting, Firefox users can now customize the keybindings for the accesskeys on any website. I’d call that progress.

(Update: The latest version of my Javascript restores compatibility with Gez’s Greasemonkey script.)

Looking into the (distant) future, XHTML 2 will offer a self-documenting reformulation of accesskeys. Instead of

which points to an RDF Schema defining the role(s). Alternatively, there’s a

<access key="c"
title="Post a Comment"
targetid="comment" />

for those instances when a custom-defined role seems like overkill.

The definition-list we’ve been talking about here is a crude stand-in for the collection of <access> elements in an XHTML 2 document. With that substitution, the same techniques for customizing keybindings will work in XHTML 2. Hopefully, that capability will be built into XHTML 2 User Agents, rather than being an afterthought, as with today’s accesskey attribute.

January 11, 2006

Dem Bones

I was admiring the “doggiesaurus” dinosaur picture my 5 year old created in preschool today, as he cheerfully explained to me about their habits (they’re plant-eaters, apparently). So I turned to him and asked, “Son, when you grow up, would you like to be a scientist, who digs up dinosaur bones?” “Daddy,” he says to me, witheringly, “You mean a palæontologist.”

“Yes,” I said meekly, “a palæontologist.” “Sure.” he says, “And then, when I’m done, I can give the bones to a museum.”

January 8, 2006

Editable Accesskeys

Accesskeys are a very nice mechanism for making your website more accessible. You can define a set of keyboard shortcuts for common navigational tasks, making it easier for visitors, who can’t (or don’t wish to) use a mouse, to find their way around your site. For instance, on this site, hitting Alt -4 (Cntrl -4 on a Mac) takes you to the search form on the current page; Alt -1 (Cntrl -1) takes you to the main page of this blog, etc.

Unfortunately, accesskeys suffer from two big drawbacks

There’s no easy way for visitors to discover what accesskeys are defined on a particular site.

The accesskeys you define may conflict with the user’s existing keyboard shortcuts. Even if they don’t, accesskey assignments vary from site to site, so there’s no “muscle memory” advantage to them.

The first problem is easy to fix. Long ago, I put a listing of the accesskeys, defined here, in the footer of each page. Scroll down to take a look at it. The markup that generated the footer is

and it can be included at the bottom of every page, using SSI, PHP or, in my case MovableType’s <MTInclude> directive. Of course, this definition-list didn’t have to be in the footer of the page. It could have been in a CSS drop-down menu at the top of the page or … The point is to make it unobtrusive, and yet easily-discoverable. Someone, with better design skills than I, could have found a more artful solution.

The second drawback seemed harder to surmount. Somehow, we need to let the user customize the accesskey assignments. But how?

Recently, Gez Lemon found a solution using server-side scripting. Unfortunately, it was more than a little cumbersome. As a web-author, it requires mucking with the markup of your pages1. As a user, it involves thoroughly unnecessary back-and-forth interaction with the server. Still, it was inspirational. Once you realize it can be done, you start to think about other ways to achieve the same effect.

I wanted something that would not involve mucking with the existing markup of my pages, and which would work client-side (so as to be as fast and smooth as possible). If you’ve read this far, you probably want to see it in action. Scroll down2 to the bottom of the page and hit the, hitherto mysterious, edit button. You can assign any keyboard character you want to an accesskey, though most browsers do not distinguish between uppercase and lowercase letters. If you want to disable one (or all) of the accesskeys, simply delete the corresponding character. When you’re happy, hit save. You can now use your new accesskey assignments anywhere on this blog. And they’ve been saved in a cookie, for your future surfing pleasure. If, instead, the current accesskey assignments suit you, just hit reset.

No markup was butchered to achieve this effect. All I needed to do was include this Javascript file and add initializeAccessKeys(); to the onload-handler for my pages. In fact, the Javascript uses the existing markup (a definition-list whose id="AccessKeyList")3 as an editing template for customizing the accesskeys.

I’ve only tested it in a few browsers (Mozilla, Safari and Opera), but expect it to work in others as well. Suggestions for improvements are welcome.

Update (1/16/2006):

Gez Lemon has crafted a Greasemonkey script to obtain similar (but not quite as good) functionality on any website with accesskeys. I’ve written a followup post about it. I’ve also revised my Javascript to play more nicely with Gez’s script.

Update (4/2/2006):

Postscript: Event Handlers

Getting keyboard navigation to work right in the presence of certain event-handlers seems to be a black art. All the sources I’ve looked at give incorrect advice. As a consequence, I, myself, have been doing it wrong for years now.

onclick="..."

by itself doesn’t work right. The user should be able to trigger this handler by hitting Return. But that doesn’t happen in Safari (and, perhaps, other browsers).

onclick="..."
onkeypress="..."

will cause the handler to be triggered by hitting any key. Which is not what you want: that breaks Tab-navigation. What does work is

Since I haven’t seen this explained anywhere4, I figured I’d spell it out here, and save someone else from tearing their hair out, too.

1 Among other drawbacks, Gez’s implementation is limited to the creation of <a href="" accesskey=""> and <label for="" accesskey=""> elements. No other attributes are allowed, and you can’t use it with <area>, <button>, <input>, <legend> or <textarea>, all of which are permitted to carry accesskey attributes.

2 You can also use keyboard navigation: hit Shift -Tab, to move to the bottom of the page, highlighting the edit button. You should be able to hit return to activate editing of the accesskeys for this site (you can navigate between the editing field using Tab and Shift -Tab).

3 The script assumes a <dl id="AccessKeyList">. If you want to give the definition-list containing the accesskeys and their meanings a different id, change the value of the accesskeylistid variable in the script.

4 Peter-Paul Koch’s otherwise-excellent table of Javascript Event support is, I’m afraid, incorrect on the status of onclick support in Safari.

There’s no getting around the fact that the Adminstration has flagrantly, unrepentantly violated the Law of the land, and vows to continue to do so. They’ve had 4 years to propose changes to FISA (beyond those of the USA Patriot Act), if they felt there was an argument (which would pass Constitutional muster) for doing so. Instead, they decided to simply ignoreFISA. Evidently, the Rule-of-Law is not on their radar screen. Everyone wants the President to have the tools necessary to fight the War on Terror. No one but a partisan hack would conclude that shredding the Constitution is the way to provide those tools.

So, where does that leave us? The current Majority in the Senate is, alas, of the opinion that Impeachment is a partisan club, to be wielded against one’s political opponents, rather than a solemn tool to ensure that the Executive adheres to the Rule-of-Law. Without tremendous public pressure, we’re unlikely to see substantive hearings, let alone Articles of Impeachment.

Perhaps a bumper sticker campaign is in order

WHO DO YOU HAVE TO BLOW TO GET A
PRESIDENT IMPEACHED AROUND HERE?!

KING GEORGE? DIDN’T WE FIGHT A REVOLUTION ABOUT THAT?

1 And engage in torture and arbitrary detention without trial and …. But let’s not stray from the current subject.

2 Arguably, by sitting on the story for over a year, the New York Times compromised our National Security by ensuring the reelection of George W. Bush. But that’s probably not what Hindraker had in mind.

January 2, 2006

CDT

After writing my two previousposts on approaches to quantum gravity, various people asked me to write something about Causal Dynamical Triangulations, a lattice model that has enjoyed a certain amount of favourable ‘buzz’ recently. I’ve been procrastinating about following up on that request because, to do a halfway decent job would require at least two post, one about generalities about lattice models of quantum gravity and one specifically about CDT. Alas, I’m not that interested in the subject, so I’ve always been able to find something else I’d rather write about …

Anyway, as a bit of New Years resolve, here’s a stab at such a post which will, alas, fall far short of what’s really required.